A conventional diatonic keyboard, such as a piano keyboard, consists of a plurality of sequential side-by-side octaves wherein each octave is formed by twelve keys with seven successive side-by-side white keys in a front row and with five black keys in a back row interspersed between rear portions of alternating groups of three and four white keys. Alternate keyboard arrangements have been proposed, such as that of U.S. Pat. No. 2,406,946 to Firestone and U.S. Pat. No. 152,726 to Cramer.
Firestone's keyboard consists of six white keys and six black keys to the octave, with a black key between rear portions of each pair of adjoining white keys. The black keys are situated above and to the rear of the principal playing surface of the white keys. The front row of keys are for the notes "D.music-flat.", "E.music-flat.", "F", "G", "A", "B" and the back row of keys are for the notes "G.music-flat.", "A.music-flat.", "B.music-flat.", "C", "D", "E".
Since the conventional keyboard has been used for centuries, it can be difficult for a pianist to switch to an alternate keyboard such as that of Firestone. The Firestone keyboard is visually and tactually confusing to a pianist already indoctrinated with the conventional keyboard. In using the Firestone keyboard the pianist requires an auxiliary reference device for identifying the keys.
Firestone also discloses a notation system including a brace of five staffs wherein the notes written on lines are played on one row of keys and notes written on spaces represent the other row of keys. According to the Firestone notation system, each staff consists of a group of five equal-spaced lines, with the spaces between the lines being equal. The lines of each group are for the notes A.music-flat., B.music-flat., C, D, and E. The spaces of each group are for the notes A, B, D.music-flat., E.music-flat.. The note G is printed in the space below the lowest line; the note F is printed in the space just above the highest line. The note G.music-flat. is printed on a semi-line equidistant between groups.
Various tunings, i.e., relative frequencies, of the twelve notes of a conventional twelve note or step per octave scale have been employed in the prior art including those known as Pythagorean intonation, just intonation and equal temperament. Table I sets forth the frequencies in Hertz (Hz) and the cents of a two octave portion of a keyboard beginning with A below middle C in equal, just and Pythagorean intonations tuned on a C scale, i.e., with C as the tonic or base note. Cents is a conventional logarithmic scale according to the equation: ##EQU1## wherein T is the fundamental frequency of the first note (in the case of Table I, A or 220 Hz) and N is the fundamental frequency of the second note. The Pythagorean and just intonations are based upon setting selective relative notes, i.e., intervals and chords, to be highly consonant. Consonance is produced by the absence of audible beats and dissonance when two notes are played simultaneously.
The Pythagorean tuning is based upon the successive setting of perfect fifth intervals and octaves between
TABLE I ______________________________________ EQUAL, JUST and PYTHAGOREAN INTONATIONS (Tuned on C scale) Cents Frequency (Hz) Pytha- NOTE Equal Just Pythagorean Equal Just gorean ______________________________________ A 220 220 220 0 0 0 B.music-flat. 233 235 232 100 112 90 B 247 248 248 200 204 204 C 262 264 261 300 316 294 C.music-sharp. 277 282 275 400 428 384 D 294 297 293 500 520 498 E.music-flat. 311 317 309 600 632 588 E 330 330 330 700 702 702 F 349 352 348 800 814 792 F.music-sharp. 370 371 366 900 906 882 G 392 396 391 1000 1018 996 A.music-flat.' 415 423 412 1100 1130 1086 A' 440 440 440 1200 1200 1200 B.music-flat.' 466 469 464 1300 1312 1290 B' 494 495 495 1400 1404 1404 C' 523 528 521 1500 1516 1494 C.music-sharp.' 554 563 549 1600 1628 1584 D' 587 594 587 1700 1720 1698 E.music-flat.' 622 634 618 1800 1832 1788 E' 659 660 660 1900 1902 1902 F' 699 704 695 2000 2014 1992 F.music-sharp.' 740 743 732 2100 2106 2082 G' 784 792 782 2200 2218 2196 A.music-flat." 831 845 424 2300 2330 2286 A" 880 880 880 2400 2400 2400 ______________________________________ keys. In the notes of a perfect fifth interval, the higher note has a fundamental frequency which is exactly 3/2 times the fundamental frequency of the lower tone so that the second harmonic of the higher note is equal to the third harmonic of the lower note to produce consonance. Beginning with the tonic, e.g., middle C in the C scale, two successive upward fifths are tuned followed by downward tuning an octave, e.g., G is tuned relative to C, D' (the prime indicates that the key is in the next higher octave) is tuned relative to G, and D is tuned relative to D'. This procedure is repeated for three more keys, e.g., A-D, E'-A, E-E' and B-E so that now the relative tuning of six keys, e.g., C, D, E, G, A and B is set. Next the base note of the next higher octave, e.g., C'-C, is tuned, and then a downward fifth is tuned, e.g., F-C'. This is followed by an upward octave, e.g., F'-F, and two downward fifths, e.g., B.music-flat.-F' and E.music-flat.-B.music-flat.. The upward octave and downward fifth tuning procedure is continued to complete the tuning of the middle octave, e.g., E.music-flat.'-E.music-flat., A.music-flat.-E.music-flat.', C.music-sharp.-A.music-flat., C.music-sharp.'-C.music-sharp. and F.music-sharp.-C.music-sharp.'. The notes in the remaining higher and lower octaves are tuned from the now tuned octave. The ratios of the fundamental frequencies of the notes in the tuned octave relative to the tonic or base note are shown in Table II for a C scale tuned in accordance with the above Pythagorean tuning procedure.
The just tuning system (also called the pure tuning system) is characterized by changing the major third interval from the Pythagorean ratio of 81/64 to the ratio of 5/4 which minimizes beats and renders the just major third interval substantially more consonant. Just tuning is initiated by tuning of the notes of three consecutive triads, e.g. in the C scale, 'F-A-C, C-E-G, and G-B'-D', and then further octave tuning these notes in upper and lower octaves. The remaining notes are tuned using the previously tuned notes, e.g., C.music-sharp. is tuned to a major third
TABLE II ______________________________________ PYTHAGOREAN & JUST RATIOS (Tuned on C scale) RATIO NOTE PYTHAGOREAN JUST ______________________________________ C 1/1 1/1 C.music-sharp. 256/243 16/15 D 9/8 9/8 E.music-flat. 32/27 6/5 E 81/64 5/4 F 4/3 4/3 F.music-sharp. 1024/729 7/5 G 3/2 3/2 A.music-flat. 128/81 8/5 A 27/16 5/3 B.music-flat. 16/9 9/5 B 143/128 15/8 ______________________________________ below F, B.music-flat. is tuned to a perfect fourth above F, E.music-flat. is tuned to a major third below G, A.music-flat. is tuned to a major third below C', and F.music-sharp. is tuned to a major third above D. Table I lists the fundamental frequencies and cents of two octaves of notes in a C scale tuned in the just system, while Table II lists the relative ratios of the fundamental frequencies in one octave of a C scale tuned in the just system.
It is noted that there are variations in the Pythagorean and just tuning systems. For example in a C scale in the just tuning system, D can be set at a ratio of 10/9, F.music-sharp. or G.music-flat. can be set at a ratio of 25/18, and A.music-flat. or G.music-sharp. can be set at a ratio of 15/16 relative to C.
The major problem with the above scales is that various intervals and chords are dissonant. The following Table III lists major intervals in cents for Pythagorean and just intonation in the C scale. In just intonation, the fifth, fourth, third and sixth intervals are all consonant with 702, 498, 386 and 884 cents, respectively, when the lower note is C, G or A.music-flat., but for the other lower notes, one or more of the intervals are dissonant. In the Pythagorean tuning system, most of the fifth and fourth intervals are consonant, but the third and sixth intervals are general dissonant even when the lower note or tonic of the interval is C. Because of this dissonance, it is standard practice to adjust or temper the tuning of various notes in the scale so as to minimize beats and dissonance in the various intervals.
The practical solution of the prior art is equal temperament wherein the fundamental frequency of each step or note is made exactly equal to 2.sup.1/12 times its immediate lower note. This equal temperament is employed in many musical instruments in use at the present time. As shown in Table I, each note is exactly 100 cents above its
TABLE III ______________________________________ MAJOR INTERVALS (Cents) (Tuned on C scale) MAJOR OR FIFTH FOURTH THIRD SIXTH TONIC JUST INTONATION ______________________________________ C (G) 702 (F) 498 (E) 386 (A') 884 G (D') 702 (C') 498 (B') 386 (E') 884 F (C') 702 (B.music-sharp.') 498 (A') 386 (D') 906 D (A') 680 (G) 498 (F.music-sharp.) 386 (B') 884 B.music-flat. (F) 702 (E.music-flat.) 520 (D) 408 (G) 906 A (E) 702 (D) 520 (C.music-sharp.) 428 (F.music-sharp.) 906 E.music-flat. (B.music-flat.') 680 (A.music-flat.') 498 (G) 386 (C') 884 E (B') 702 (A') 498 (G.music-sharp.) 428 (C.music-sharp.') 926 A.music-flat. (E.music-flat.) 702 (D.music-flat.) 498 (C) 386 (F) 884 B (F.music-sharp.) 702 (E) 498 (D.music-sharp.) 428 (G.music-sharp.) 926 D.music-flat. (A.music-flat.') 702 (G.music-flat.) 478 (F) 386 (B.music-flat.') 884 F.music-sharp. (C.music-sharp.') 722 (B') 498 1 (A.music-sharp.') 406 (D.music-sharp.') 926 PYTHAGOREAN INTONATION C (G) 702 (F) 498 (E) 408 (A') 906 G (D') 702 (C') 498 (B') 408 (E') 906 F (C') 702 (B.music-flat.') 498 (A') 408 (D') 906 D (A') 702 (G) 498 (A.music-sharp.') 384 (B') 906 B.music-flat. (F) 702 (E.music-flat.) 498 (D) 408 (G) 906 A (E) 702 (D) 498 (C.music-sharp.) 384 (F.music-sharp.) 882 E.music-flat. (B') 702 (A') 498 (G.music-sharp.) 408 (C.music-sharp.') 906 A.music-flat. (E.music-flat.) 702 (D.music-flat.) 498 (C) 408 (F) 906 B (F.music-sharp.) 628 (E) 498 (D.music-sharp.) 384 (G.music-sharp.) 882 D.music-flat. (A.music-flat.') 702 (G.music-flat.) 498 (F) 408 (B.music-flat.') 906 F.music-sharp. (C.music-sharp.') 702 (B') 522 (A.music-sharp.') 408 (D.music-sharp.') 906 EQUAL TEMPERAMENT ALL 700 500 400 900 ______________________________________ immediate lower note. As shown in Table III, all the fifth and fourth intervals are generally consonant since noticeable beating and dissonance doesn't begin to occur until the interval varies more than about 4 cents from a perfect tuned interval at notes in the middle octaves. However, the major third and sixth intervals are dissonant in even temperament to cause triads or chords to be somewhat dissonant since each chord includes both major and minor third intervals along with the dominant or fifth interval.
It is an object of the present invention to provide a musical notation and keyboard arrangement system which is advantageous and conducive to musical education.
It is a further object of the present invention to provide tuning for a musical instrument which is highly consonant for all major intervals and chords.